While solving a mathematical problem, Samidha squared a number and then subtracted 25 from it rather than the required i.e., first subtracting 25 from the number and then squaring it. But she got the right answer. What was the given number?

### Concept & Formula Translate the word problem into an algebraic equation. Let the given number be $x$. * Samidha's operation: $x^2 - 25$ * Required operation: $(x - 25)^2$ Set them equal to find $x$: $$x^2 - 25 = (x - 25)^2$$ ### Step-by-Step Solution * **Formulate the equation:** $$x^2 - 25 = (x - 25)^2$$ * **Expand the right side** using $(a-b)^2 = a^2 - 2ab + b^2$: $$x^2 - 25 = x^2 - 50x + 625$$ * **Simplify and solve:** Subtract $x^2$ from both sides: $$-25 = -50x + 625$$ $$50x = 625 + 25$$ $$50x = 650$$ $$x = \frac{650}{50} = 13$$ ### Exam Strategy & Shortcut **Option Verification Method:** Test the options directly into both scenarios. * Try Option (a) $13$: * Scenario 1: $13^2 - 25 = 169 - 25 = 144$ * Scenario 2: $(13 - 25)^2 = (-12)^2 = 144$ Both values match perfectly, confirming $13$ is the correct answer within seconds. ### Common Pitfall * **Mistake:** Thinking that since a negative number arises inside the square, i.e., $(13-25) = -12$, the method is invalid or results in "Cannot be determined". * **Correction:** Remember that the square of a negative number is always positive: $(-12)^2 = 144$. ### Final Answer **Therefore, the correct answer is 13.**